Chapter 4.3, Problem 29E

(a)

To determine

To Show: The differentiable inverse of the given function.

Explanation of Solution

Given information: Given function $f(x)=\cos x+3x$

Formula used: $\frac{d(x^n)}{dx}=n{x}^{n-1}\text{and}\frac{d(\cos x)}{dx}=-\sin x$ with addition rule.

Calculation:

  $\begin{array}{l}f(x)=\cos x+3x\\ \text{Differentiate}\text{with}\text{respect}\text{to}x\\ f\text{'}(x)=\frac{d(\cos x)}{dx}+3\times \frac{d(x)}{dx}\\ f\text{'}(x)=-\sin x+3\end{array}$

Since $f\text{'}(x)$ is defined for all values of x.

Thus, $f$ has a differentiable inverse.

(b)

To determine

To Find: The value of $f(0)\text{and}f\text{'}(0)$ of the given function.

Answer to Problem 29E

  $f(0)=1\text{and}f\text{'}(0)=3$

Explanation of Solution

Given information: Given function $f(x)=\cos x+3x$

Calculation:

  $\begin{array}{l}f(x)=\cos x+3x\\ Differentiatewithrespecttox\\ f\text{'}(x)=\frac{d(\cos x)}{dx}+3\times \frac{d(x)}{dx}\\ f\text{'}(x)=-\sin x+3\end{array}$

  $\begin{array}{l}f(0)=\cos 0+3\times 0\\ f(0)=1\\ \text{and}\\ f\text{'}(0)=-\sin 0+3\\ f\text{'}(0)=3\end{array}$

(c)

To determine

To Find: The value of $f^{-1}(1)\text{and}(f^{-1})\text{'}(1)$ of the given function.

Answer to Problem 29E

slope is defined for all values of $x\text{and}y$ except when $x=0\text{and}x=2y$

Explanation of Solution

Given information: Given function $f(x)=\cos x+3x$

Formula used: $\frac{d(x^n)}{dx}=n{x}^{n-1},\frac{d(\cos x)}{dx}=-\sin x\text{and}(f^{-1})\text{'}(x)=\frac{1}{f\text{'}(f^{-1}(x))}$ with addition rule.

Calculation:

  $\begin{array}{l}f(x)=\cos x+3x\\ \text{Differentiate}\text{with}\text{respect}\text{to}x\\ f\text{'}(x)=\frac{d(\cos x)}{dx}+3\times \frac{d(x)}{dx}\\ f\text{'}(x)=-\sin x+3\\ f\text{'}(x)>0\end{array}$

Now,

  $\begin{array}{l}f(0)=1\\ f^{-1}(f(0))=f^{-1}(1)\\ \text{Thus,}{f}^{-1}(1)=0\end{array}$

Also,

  $\begin{array}{l}(f^{-1})\text{'}(1)=\frac{1}{f\text{'}(f^{-1}(1))}\\ (f^{-1})\text{'}(1)=\frac{1}{f\text{'}(0)}\\ (f^{-1})\text{'}(1)=\frac{1}{-\sin 0+3}\\ (f^{-1})\text{'}(1)=\frac{1}{3}\end{array}$